Discussion
Let \(\{X_{n}, n \ge 1\}\) be a sequence of random variables such that \(P\left[\lim_{n\to\infty} X_{n} = X\right] = 1\) for some r.v. X. Then which of the following is not always correct ?
Answer: D
The statement not always correct is \(\lim E[\exp(X_{n})] = E[\exp(X)]\).
Almost sure convergence carries over to continuous functions: \(\exp(X_{n}) \to \exp(X)\) a.s. (C). For bounded continuous functions such as \(\cos\) and \(e^{itx}\), the dominated convergence theorem then gives convergence of expectations (A, B).
\(\exp\) is unbounded, so expectations need not converge. For example, on [0, 1] with Lebesgue measure let \(X_{n} = \log n\) on \([0, 1/n]\) and 0 elsewhere. Then \(X_{n} \to 0\) a.s., but \(E[e^{X_{n}}] = n \cdot \tfrac{1}{n} + (1 - \tfrac{1}{n}) \to 2 \ne e^{0} = 1\).
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